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COSC 4364 Numerical Methods - Lecture21_Eigenanalysis_SVD
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00:00 gorgeous thing. Mm hmm. Transcript hope. Okay. So in your
00:15 . Oh man. I didn't get talk about one of the last spicy
00:22 it's called triangle subspace methods. And are not covered from the book as
00:27 as I remember. And I'm not to go through a little detail.
00:32 just wanted to effect will be aware these methods exist because they are very
00:38 used to practice. But there are advanced. So we're probably doing more
00:43 that of course that And in his for life after class, it's supposed
00:51 be at least aware of the And yes. And then,
00:59 most of them today when we're dealing this so called singular value decomposition.
01:06 . So here is kind of, , again borrows license. It's not
01:11 problem in the book. So, , you can, since we talked
01:18 the farm method and that was essentially all these guys until and I'm using
01:26 then your functional to figure out the values the Lord is the magnitude
01:30 So this is not quite the way that actually work competition early. But
01:35 is sort of a matter of places the rest of it here. This
01:38 tell us what you're actually going to . And I just wanted to
01:46 I guess this step here and these here so we can use this kind
01:50 subsidies methods. Then what ends up again is fewer characterization of these matrices
01:59 they actually don't compute these guys and show you on the next side.
02:03 huh. And so I tried to the last time that you were
02:08 if you are a characterization is the common in this method when it comes
02:14 a country with high value. It's involved. They find credible subspace
02:21 And so you can plug in and at the respective position and you look
02:27 this expression and after some manipulation, can figure out how this structure,
02:32 happens. And then from there this apart december this time around. It's
02:42 householder, it's this bad. Um materials done the effective as a complication
02:50 procedure. How you get they suggested . Okay. And eventually yeah,
03:02 of and your values at first. as it says here someone you have
03:07 use a different procedure to get the values. So everyone hands up in
03:14 particular form or disappear. That the of the details can figure out how
03:18 deal with this guy and get approximations he ends up being something known as
03:28 that is commonly used. And they'll back when they're commonly used. They
03:33 commonly used when they into sports So the point of these methods are
03:41 the computational intensive parties again, matrix multiplication. So that's pretty much only
03:48 one thing to expect from application essentially each declaration steps and trying to
03:54 I guess so that's the one thing remember that this crisis and subsequent writes
04:03 it. That's something you want to at is sportsman disease and then they
04:12 particularly useful again because the computational the part is making expected multiplication, but
04:17 an adjective as far as this is too much. So there's a little
04:24 more I want to do to get things. As I said, you
04:27 get exactly what you want, but are close enough that usually and we'll
04:34 back to that. They didn't talk . It's a method for solving systems
04:37 equations, but um the reason why has come later in the book is
04:44 I understand how those things work depends understanding, not just from interest and
04:52 rights of this. And then there another version is used for symmetric majors
04:59 . Angeles for this. Essential the here. Well, thanks again,
05:04 its like for stuff and they have little bit different, compute the next
05:12 , but that's pretty much what I to say in a little bit.
05:15 also one that may not do so , but again, I don't expect
05:21 okay. Um no plan is not no, that when these crowd of
05:28 methods might be useful things get into of exactly how all the steps are
05:36 get some rest. So it's just principle you can and that's why these
05:43 methods and we'll come back and talk it if that they're solving systems of
05:48 is that the point is many times seems converges quickly, then you don't
05:55 too many interesting. Um but the was direct followers of the cult about
06:04 in the classes. Pretty much they know much until they have solved the
06:10 problem. There is no step was of fundamental. You don't know and
06:14 think that's the best I have to through the whole procedure, but that
06:19 , which is not true. That's . Okay, so this is more
06:26 on the same. Um that's that's this is what I just wanted to
06:33 and we're aware of that and they at both solvers after the packages you
06:38 find kind of substance some overseas. is just to make the world if
06:44 were forced. All right, so little bit of condition numbers. So
06:55 then we'll come back to, I to get to this point today also
07:02 they're used putting other things now this time has to do one tradition numbers
07:08 computing again, back to this. what this sign basically talks about as
07:20 , you have some errors for in the making you want to compute
07:26 onion values. So parece may simply rondo fairer stuff that's whatever before.
07:32 you got the a horse or however got computed your a is not exactly
07:37 you wanted it to be. So . That's some terrible things and then
07:42 the question of this, this area effect, the union values and what
07:49 on this filling the stylists and they at some normal there in this case
07:55 used to normal. So you get of the size of what magnitude or
08:01 and this is the number, it's norm and that defects are your
08:07 Them depends on the condition number of matrix A matrix that is the matrix
08:15 my investors. Well then majors you know the investors from the
08:22 then a particular condition number. And you can basically figure out how much
08:28 the reel I can value for this A as they computed, I guess
08:35 with they're somewhat their own use many . So it feels to how best
08:42 the error in the computing on the of value that is unrelated to the
08:48 number for the make it so I vectors and what about coming from this
08:56 that shows what the foundation number is it's related to the ratio between ah
09:07 a lot of investment in the ratio united bodies on the matrix And the
09:11 of west of it Ratio between the and smallest one. And I'll come
09:17 after that said concrete examples that it's again, it's always good as I
09:26 to say that it has an announcement the class and none of it is
09:30 to get to a sensible how much you trust the outcome or whatever you
09:36 . So this is again that kind way of trying to see how much
09:40 and certain things can influence the Alright, in summary has something to
09:48 to step forward. We know by , hopefully this is the argument of
09:52 problem. General various properties here, organizations and things are really have some
10:01 um what the island writers should Not so much about the I am
10:07 and then they used this record cash in this in some ways it was
10:12 . At least sometimes you can guarantee the simple things are doing the absolute
10:18 of relevant columns except for the angle that you can see if there's a
10:24 weapon. Matrix is singular or you guarantee itself. So it's kind of
10:29 very simple procedure and then he said , the book come to the most
10:35 on this thing which is formally correct it's not the different useful in Iraq
10:41 and have breakfast. And then I to point out what is the official
10:47 what to choose for methods in terms computing and I have directors and is
10:56 long list that we want to very but I try to I like the
11:03 is of particular rather than saying wow situation. So let's get your son
11:11 not only for final exams but you know remember things you you're very
11:21 don't come in. Alright. So was it about imagine having breakfast
11:32 Mm hmm it's not. Um, anyone knows about STD for short and
11:43 value decomposition. What happened? I , they finally see the definition.
11:50 , I will come up. they will come up. So it's
11:57 heavily used. I would say it's extension or refinement of iron values computations
12:06 I'll talk a little bit better about system. Enough. Let's see what's
12:12 . I think they decided to That's a little bit of a different
12:15 of things to show another video. one is better quality and it's not
12:21 by me or is one of my . Nothing. Okay. So
12:37 man, trying to figure out what has to do. You think about
12:42 ? The composition? But it And the so as we can see
12:51 are drones, Let's come on nine but small ones fire because like somebody
13:07 that's sitting there basically discover the environment . Something that, but it's not
13:16 good that they have to crash into trees. So we have to find
13:20 own way and on the streets. we basically have cameras look at the
13:27 , something else along and on campus they're able to. So, and
13:33 we're trying to stick it out again environment and not the navigates, but
13:40 not somebody else looking at the So this looks awesome. So that
13:45 a better quality of the supreme So that's true. So I'll come
13:52 to what that has to do with . Can you provide that? The
13:56 is just one type of application but used that often the graphics and
14:05 So now I will be kind of to this. What were the reminder
14:09 what I can value? The semi were about? So this is yeah
14:16 doing it and I'm sorry, do have an introduction? So let's first
14:21 you know, as we know from compactor is such that the naked state
14:27 gives it and normally it also kind rotates it. So that's what it
14:34 . Something that maybe it's been in coordinate space and it turns it into
14:38 . It's so easy the substance of hoped for rotated and scaled and
14:45 So I love this. It's good on the way to trying to understand
14:50 value decomposition. The first kind of to look at more distorted the idea
14:57 records and Eigen values and form something I want to say, I can't
15:01 composition. So in the again, composition that mr matrix cannot be expressed
15:10 this form of a product where the is the metrics of dynamic factors and
15:19 is this diagonal matrix and all Eigen . So basically you can factor or
15:26 composed the matrix A and this particular , I will show later why this
15:31 kind of a useful form. But this is um they find are moved
15:37 just finding onion values and I am and also trying to get them use
15:44 rest of the composition with politics. the next slide said that if you
15:53 that these are the Eigen vectors then . Say basically yes, scales it
16:00 of the factors ah the item if you're fine and you're the matrix
16:08 it. That's back on to the safe, safe, safe. But
16:14 escape. So that's when you see coordinates um of the direction of the
16:20 vectors is the same but they get . And so this is just if
16:23 work on again, factors corresponding to cavity, what you hear?
16:33 what's the subject normal is there are things happened. Not only does things
16:37 scaled but they also get rotated. then there's this other decomposition is not
16:42 as the singular value decomposition foreigners that's kind of the same structure about two
16:50 legacies you leave known as singular vectors mm hmm right, singular directors and
17:03 is still a diagonal matrix but it's no longer. And the objects are
17:09 get values. It's known as singular since I guess what I think
17:16 So this is kind of the singular decomposition and all the matrix states.
17:24 we'll talk about giving examples why this useful. Um Thanks to no a
17:33 value decomposition. It wasn't he was much until too long ago in terms
17:43 mathematics because people didn't have a good of computing you said. So it
17:52 still computational and potentially expensive, but was too hard to nobody. It
17:58 take off until finished 30 plus years when just a long time for you
18:06 the math context. So No one then look at this form eight
18:14 He looks like the old thing you know, and I think many
18:19 his collection of mayan vectors. Um now it's them. Then we have
18:28 hmm. Making so I get values another set of. Um So,
18:36 so the so the B one the basis, both of them, both
18:43 them generally all the same dimensional. it turns out that the single value
19:00 the composition exists for in the you know, regardless whether it's symmetric
19:06 not, whether it's real or imaginary complex or whether it's square or rectangular
19:12 goes. So it's that's points. when the two vectors from now we
19:20 about say this for spirits and it turns out that singular value decomposition
19:27 good for your feelings. Um, but different, you know, unravel
19:33 sort of major expectation and goes back normal vectors. You can see it
19:37 very much like the I value highly things except now I'm going to work
19:44 two different sets of nominal basis. the baby and they do and we
19:50 the singular values instead of the island unraveling this Singapore responded and adjustment.
19:58 see that that was the first. second problem is the lights inside looks
20:05 the difference in terms of the general of relations. But of course if
20:12 have it but I can value their door history. That may be
20:24 Some famous thanks probably for another investing another one. But anywhere you can
20:31 that this method has been around for some time but we never caught on
20:38 , you know, 100 plus years . And here is the future of
20:42 guys that uh, is useful for make something a little bit. I
20:48 a tangible sense of MR Burton is colleague and friend. Um, that
20:57 quite the good fight mathematician used to an Oxford University. I have been
21:06 of popularizing its speed and but among is this fellow jean gold and also
21:14 at stanford and professor correctly, they the ones that basically and they're figuring
21:20 of the computer single value of their . Oh and don't worry about it
21:29 . So, and then as you , Mr um has also been an
21:35 contributor and software but also comes from . Mm hmm. The singular value
21:44 and various the one that has contributed the problem the most the strongest contribution
21:54 on this. Professor jean Bolen at produced driving on the cargo um,
22:02 that service in order to consult a bit history. It's not. Yeah
22:11 . Um So it's a certain when talk about the squares conducted by the
22:20 and other things called so the universe covered a little bit today so the
22:26 to make this a singular. Um that's right. For those who don't
22:32 soundtrack to compute the inverse it's not . And of course it's a singular
22:38 the universal respect. So too the is is a way of dealing with
22:44 that are singular and we'll put it a sensible way. I understand what
22:49 getting. Okay so not more motivating not just flying jobs so many questions
23:02 alright. So from and other things mathematicians build the strength of the best
23:10 and he's written a couple of very books on the application. So there's
23:18 an example as effectively using STD for compression but it's not talking with dana
23:29 but data compression. So in this I think an image that has a
23:37 of pixels and each pixel and in simplest case it's just the great scale
23:42 that must have one number that they At the particular big sell so 2,005,000
23:50 Image done the best this way they the civil matrix 1008019th century that intensity
24:01 So to do it good for us this context. And I said you
24:05 you want to trust me. You all the pixel values. That's kind
24:10 the brute force approach. Now if just thinking about their composition they're not
24:16 to do that. And so one take this matrix A. And then
24:22 pictures the singular invited the composition of matrix. Um So I take a
24:28 at that. I remember there was few times it's matrix sigma times on
24:36 right hand side the be transformed. are just so if you kind of
24:42 it what reaction you got this collection Americana? Common vector trans director and
24:49 with the first single value. And we take the next and two vectors
24:53 from you and one from B. so this is a collection of other
24:58 with the corresponding singular values. So maybe I actually don't think that's
25:07 good job. So that's basically or from Mr so the unrivaled distance what
25:16 see Is that? Yes This one this one then this summer because the
25:26 of. Oh so here is where were. So now then the next
25:38 is not. Yes. Yes. here is kind of taking it the
25:44 that the inmates and so this is smaller and just started to like opportunity
25:51 the principle. So so this defective values for this image. So I
25:58 to think about having to matrix and you start to compute the median values
26:04 sort of single values do the STD it. Senegal is given the lead
26:12 the sequence in softer eight of them with their corresponding singular value. So
26:19 first one looks like this doesn't look much like this guy. Ah Then
26:25 take the next two and the next and the next two and none of
26:29 of the Very much like this one should put them all together so I
26:36 superimpose them. And as you can with this eight singular values and their
26:46 and directors then this guy he uses of the inspectors doesn't look too badly
26:57 to this one. Mhm. So think the next. Yes. So
27:05 it says here so in this case just computed the difference. Right?
27:10 there was thousands and 1000 matrix. a million pixels. So instead here
27:16 one works Stop after 60 of that's still the family dollars number.
27:21 it's fun Then you have 66 months it's not too much. But the
27:25 are still The same dimension as the so that there's 1000 elements in the
27:33 . So in this case if you of them there is kind of a
27:37 . What a production it's not but not but it's kind of the compression
27:46 effective compression, let's just try to this. Hungarian complete example that things
27:53 become fairly good enough to in many if you want to do is supporting
27:59 process. So this is more related the video of the jobs. So
28:09 there's been something that is called the single localization and not being a slam
28:16 short. So trying um the device or something else ah looks at the
28:30 and based on what it sees is to figure out where is it with
28:34 to the environment. And then of there are other things that are staying
28:39 , navigation. So that's what's known localization, where I am I and
28:43 also the bad thing on environment. you know where you are feeling.
28:50 the SPD is in fact one of workhorses and trying to deal with
28:55 I'll try to give it in terms the china example. So, um
29:01 of these drones, so one set sensing the environment is he stepped off
29:13 cameras that has been measured the Giving me one time. So,
29:20 one particular sensor that is used as as it's lighter. Best adviser,
29:27 , electrical perfection. Um, so do you measure the time between the
29:33 and the reflection things are good So even with the, We went
29:39 to 50 seconds, you can actually that consideration then reducing the speed of
29:44 to figure out as far as whatever reflected. So your best finger,
29:52 bunch of points. Um, this a 0.12 and nowadays that's used for
29:59 some code and stuff from videos. particular characteristics and environment and used its
30:09 . So you get this point class of course this houses or sent out
30:18 . So you have to realize if look at Later liner's gap a few
30:25 here in the millions of points per . So it's the serious and all
30:32 data that comes in to trying to sense of and of course take the
30:37 an example rarely is it sitting in place and taking our pictures to get
30:43 better assessments that moves around. So looks like the same thing from many
30:48 angles. And somehow then you need figure out how to emerge is different
30:54 such effective against point clouds from different . We expect some environment and that's
31:01 kind of this review comes into play areas Australian examples where there are assisted
31:09 face shape and they may have uh taking several points or whatever observations.
31:20 from two different in this case. This structure is rotated and translated absolutely
31:30 . And they also have a different in this case is also different
31:35 But we're supposed to assume that the transformation of the same objects. But
31:44 you have this set of points and have this set of points and they
31:47 to figure out are these in fact of the same structure or is it
31:53 somebody during conciliation. We'll figure it if these things representative, same thing
31:58 discrimination, translation. So if it's rigid body, then it's simple.
32:03 know that it's supposed to be rotation translation. So there is best attractive
32:11 between the points and the pointers. I'm trying to figure out how to
32:17 this. I have nothing. So is kind of them when you have
32:22 some importation and translations of, let's them on and you can visually do
32:31 . Not too bad. But this and there's something in in general,
32:36 , you can't get an exact much a different story but in the sense
32:40 when you try to minimize mm hmm between the concept and that's a whole
32:47 of computation and yes, slam procedure because in real life and the globe
32:55 then there's some obstructions that you don't the same thing and the most about
32:59 points and extra points. Yeah, second. And anyway, so this
33:06 kind of what fun ends up doing went to the shoulder. So basically
33:10 have these two points, it's in this case and since it was
33:18 in this example to be rigid body it and rotated them. So and
33:27 so with where it's not not going go into those more documentation than the
33:33 and uploaded. But um, the is simple that on us What one
33:40 is that comes on the next slide guess. But first is if you
33:46 out the center of mass from the past and now has the same
33:52 the invisible, even though it should this um same central mass. Of
34:03 if you don't get the exact same points that cannot keep the truth from
34:07 central point. But forget about Now the question is doing it so
34:12 then your best is going to try move them. So they have the
34:15 sense from us and what's left is figure out the rotation. So this
34:21 what's going to be on. And was a relief for the masses.
34:24 is just since supposed to be the center of mass after the rotation and
34:29 of this place here is the original that can measure and you rotate them
34:33 then translated. And if you've done right, then the central mass of
34:38 guy is the same as the central of the other. And that makes
34:43 manipulated expression. And this is a transformation. So these things community and
34:49 out the rotation and then we have central mass as it was for this
34:53 point, Kraft said. So then need to figure out they are the
34:59 and I just figured out the rotation it's easy to figure out the
35:05 So now skipping a bunch of steps . So this is first it's the
35:13 side to ah the it could be from one to the other. But
35:18 simple isn't movies, one of them set the origin of the coordinate
35:22 So in fact the original ones, know what the center of mass is
35:27 , where it's located and then shifted the origin of the important systems and
35:33 and others you don't necessarily know Well, it's this shape that you
35:38 and then you have how the shifting each one of it points is.
35:46 these two crimes and crimes are And then you can form this matrix
35:51 you know what this makes sense. from the sites on the web,
35:54 the rest of this is now this a vector set, the coordinates for
36:00 point. And so it's this one this is not a row vectors of
36:04 , this and this and managers. . You know, So there are
36:10 this platform now the singular value decomposition this matrix age. And then actually
36:17 can find out the proper rotation, your best hit. Yes, that's
36:22 this product of the the reforms of vectors that comes out this singular
36:28 Yes. And then you just plug in. So it's a bit of
36:34 to get that. But it's just to point that this was an example
36:38 show that singular value decomposition place is of the methods and it's often one
36:43 the better methods to figure out what translation and rotation it should be on
36:49 point. Mm hmm. Yes. sure that there's successful samples of the
36:58 represents the same environment. Okay. questions on that? You know,
37:09 a few more examples here. All the singular value decomposition. I think
37:14 time they said that one of the for Medusa is we want the dimensionality
37:24 . Um, so 20 years, simple example. They have a bunch
37:30 data points and as we can see kind of one dimensional in some sense
37:38 there are more or less falling on red line now that the world is
37:43 there and gets far up the some distance off. Lord said there
37:47 investments following his heart. So if were all the plane here and that
37:53 be different. But as long as tried, they collected around the red
37:58 and may be sufficient for some You have to represent the data set
38:03 them. And there's a little bit complex example when the three D Datasets
38:10 it turns out most of the points the dataset actually falls on the flight
38:14 it's not the plan and you parallel two of the access or aligned
38:20 those normal is aligned aligned of the . So what single rather their composition
38:28 can I help you? And so this case to figure out what the
38:33 set the coordinates taxes is for representing data sets. Such them the ones
38:39 are not following in the plane. private coastal, wow. Uh concrete
38:51 for stanford guys. So it just at whatever something a bunch of customers
38:57 I don't want to hear you different of the week here and whatever the
39:02 means they don't want you talking to or doing something for them. But
39:07 best for parents, you know, in fact for the first three or
39:11 guys, best when they use whatever friday and or the other three bites
39:18 in the last one standing is the on Saturdays and Sundays. So here
39:24 the only instance what? Seven by but for metrics. So it's
39:31 Um And yeah. Environment just but you can see that in fact
39:40 represent each one here. Either the of this vector or scaling of
39:48 So basically mr So in the plane by these two factors that are in
39:54 case you can actually represent all the in the table hip optical efficiency to
40:04 left. Yeah. Okay. Nothing talking about right? Um remember.
40:15 , we can't even define it as so if you look at the columns
40:20 the matrix and you can see you see whether how many of them are
40:26 independent from each other. Right So that is the number of indian
40:34 . That's exactly this kind of the . So so it turns out I
40:39 that same level of their composition is good metric, correct? I'll come
40:46 . So that's the roles are Ah So No, it was around
40:53 and 1 thing figure out so Why? Either way? Mm
41:04 So you can do the third row equal to the first row minus the
41:13 row, correct. So the third is that in the combination of the
41:17 two rows? The matrix is Uh so it's not round three.
41:27 zero for that matter but it has used to and then when I for
41:37 response um so this is what I . So this is I want to
41:43 stuff again too. The views and nationality of the problem. So The
41:49 paper example that was seven x Used by kids The two factors.
41:57 there is another one going to come . Yeah. So this is what
42:01 matrix here that in terms of except business sector is the first one and
42:07 second one, they are not normalized we do that. We can write
42:12 the third vector as a linear Are they still? So I guess
42:19 this low is that square? Ah coefficients for these two basis vector.
42:26 aligned with This factor and has zero . That's correct. And see where
42:32 next row. It has zero components the first direction and exactly one
42:39 And on the steps And that's what said. The 3rd 1 is it
42:44 your combination of these To subtract from second one from the obviously reduce it
42:55 one of the three dimensional problems. two dimensional problems. And in this
43:00 is kind of exact there's more error the come on. Okay. And
43:08 is just right. Just talked through . So that's not. And it
43:14 that all the time there have been several and this stress that that's the
43:25 of um getting good many behavioral. . I think also there's additional separate
43:36 and I don't think so talk about it's also wanted to address for component
43:43 that it's a common and um has really good for your home and finding
43:48 principal components in the has begun related this thing. So that's defining the
43:57 momentum's or access in the dataset Most of what's your name Now?
44:05 is 10 cents a popular thing in of reference. Ah Samantha economics is
44:14 wanna some of our talk to the are at some point. It's interesting
44:21 analysis. Again, not similarity, . Mhm. Again, single breasted
44:29 composition and his violence and what methods use. And you want to figure
44:35 out the similarity and dissimilarity. And this case come back to
44:50 It shows that the the nationality of factors or maybe just historic um are
45:00 the same soul, economical things. everything is from the square. That
45:07 must be based on this case. a lot more roles than columns and
45:19 matrix. It is square and another Roosevelt. It is for all the
45:27 algebra has to do the same government the columns here about the number of
45:35 of this guy does not necessarily to spectrum of application. I'll give you
45:42 but let's just chose that and I've a lot more than how it gets
45:47 decide on how much you needed on in order to do this dimensionality.
45:57 this is just kind of not trying then you have to do this thing
46:01 this is the highlight and in south and as cities excuse to do effective
46:08 reduction and it tells you everything. the singular vectors are very closely starting
46:19 parallel target, all the documents are and if there are almost marginal than
46:30 . Okay, in the form of this is just going to give you
46:37 rent on application where usually they So now a little bit more on
46:48 formal description. That's right. Um yes, we did exist for any
47:03 . Yeah. So um now it's square and feet and then this possibly
47:16 value decomposition. So on this is of the americans, you know,
47:22 and things like that. So um and it's just a diagonal, it
47:28 like a band but it's not, just the collection of the signal
47:37 the result of statements on the same and it's called the left singular
47:43 The new guy is the left of sigma and the feathers on the right
47:47 the right singular vectors. That's our . Um, and mm hmm.
47:59 I remember sigma. So, you can do that here. And
48:12 the president says is since these are know I think of it is the
48:19 by their values and this will be baptist. So this part here is
48:24 thing that where the vectors maps onto . Just kidding. So that's what
48:33 call these are the access. These the things that comes from that tells
48:40 on this kind of investment. This working ah now and when you start
48:48 the right single values But they are ones that then gets affected on.
48:53 this against three images to kind of of it. They use are the
49:00 that kind of corresponds to. I trusted composition. I've never mm
49:14 Now. And the books they're Yes. Yeah, maybe. Thank
49:27 . But they're not necessarily make so times. Not many of these sites
49:34 been used for when we're using explore this formulation. So the government implemented
49:45 then it's okay to also have a of it for you. You don't
49:52 to fill it out. Um, the other bunch of columns. The
50:00 on the Swingman metrics houses and the . Thanks for having a so because
50:09 this um history panel a matrix It's principle only has as many non-0 singular
50:19 as the rise or that's the number in their independent columns in bed.
50:28 that's the board to kind of fill out. That means These are definitely
50:33 to be zero. Some of the values from this part may also be
50:40 . If a yeah is um now cannot have a more independent in their
50:50 , vectors than the number of But if they all vendor independent then
50:57 have as many non zero singular That's mental problems. So in some
51:04 the number of rolls underneath me is to the grant of this major
51:13 So the reduced one doesn't understand it have kind of allow for the grants
51:21 to the number of roles. So means ah yes. You know these
51:30 that um the new proceeding, it this used for you because it kind
51:38 all those the number of and principal values. So and I'll show you
51:46 example and the need to create additional vectors. There are now not necessarily
51:54 from the confidential they are not from . So it's good to know that
52:02 be some questions what the difference is the reduced the city and the so
52:12 , now coming back to something we about or um so any questions on
52:25 things are probably full of course, to use the singular values. So
52:35 . So we'll have, you your rectangular matrix one can make it
52:42 kind of thing. Was in a aid that becomes kind of an N
52:46 N matrix. And depending upon the mm kind of large, it's kind
52:54 like another brother or maybe a product kind of the smallest one.
53:01 and some of the straightforward methods for power, least squares fit within the
53:08 . Some of those methods tend to this matrix, but it's usually a
53:12 idea. But no matter. But know, that's that's a useful thing
53:19 you discuss a little bit relationship between , singular values and by your values
53:31 agents. So and again, the there at some point back and the
53:39 matrix permission, which is called it's symmetric is such as and transport it
53:47 get the facts and your transports can get they also get it back.
53:51 it is information and then there was and regarding us through the similarity transformation
54:03 and the diagonal matrix tube. And that case we know it's similar to
54:08 transformation by the finding two. That's right on this matrix. And in
54:15 section gets there over this way. you can kind of recreate the similarity
54:20 , I guess the diagonal matrix multiplication the right for a few and from
54:26 left to inverse. And then they to inverse peoples of the world.
54:32 it's just different when you're right. than this Is in fact I can
54:48 of 88. Not to me. . This is not the natives.
54:53 this is the gun violence at this . And as I turned it down
54:59 you, I am directors uh you do it multiplication from the right
55:12 And that's right witness these scales is of the columns here. So I
55:19 the correspondent problem here that means the of to our investors Of 80 a
55:27 to that. I know that this be about them. That's and then
55:35 next year on the bottom that shows it's an escape preserving because what from
55:43 nature it's not I guess I should normal but also normalized. So there's
55:53 try to look at the scale of Anthony A T A X. Yes
56:02 might divide and say, okay look this. This is this is the
56:05 here. A T X is It's almost very insensitive. Obviously I'm
56:18 and Oh yes yes. Oh so now so in this context Lando's are
56:37 ideal values of A. T. . So now someone could take and
56:49 the matrix and it's transposed at the time margins so obviously these guys are
56:58 of squares. Oh yeah I can and so that the singular values that
57:06 incorrect. Maybe that's supposed to be room On the island values of
57:14 And those are the unnoted and assistant and I guess this is whatever they
57:21 about the inside that you plug it and then you can see that the
57:28 of Q. Are the tiny electrodes 20 and calls for me mm
57:36 Correspondent diagonal or state now. And is not coming back to why I
57:47 that this is a good way of out the rank of the natures because
57:55 you sort them anonymous sources, singular at some point is there are some
58:06 are zero and that means that the Is equal to the number on 90
58:16 dinosaur. I am starting. So general if it's a full rank and
58:24 the rest of R is equal for . But in many cases that's not
58:32 . So that's why I said expensive do the same that in their
58:39 But in principle you can find the . Federal agents from the singular value
58:46 their composition. So this is what deserve right? Some of the
58:54 It's number one singular values. So , especially in practice you may choose
59:07 if the singular risers are sufficiently small the addition means in a relative
59:14 um, so expensive. The reason spread and the values um, relatively
59:22 largest at some point in the sort and decreasing magnitude you may say that
59:29 small enough. So I don't think tender comes in, so that's what
59:34 call it an american bank, there's to do, kind of an
59:40 And if they get sufficiently small then decide. And in the northern colonies
59:48 to have reduced the rank. That the dimensionality of the problem. So
59:57 is part of doing the um dimensionality . You figure out practically many of
60:06 singular values are large enough that they to worry about the corresponding so the
60:12 for singular actors and postponing space. then the capital things in a vector
60:19 and it's a lot smaller and potentially you're getting. So what's the point
60:30 this concept in America? Right. the next thing that system was to
60:43 they have a singular value decomposition and inverse is pretty well defined as an
60:52 as it's not singular. So um the best of what we get
61:01 is the correspondent single by the competition the inverse of an interesting. Now
61:11 is singular. A bunch of the months are zero. So as we
61:20 , the lighting was zero is not good thing. So this doesn't make
61:25 in case it is a similar you seem to make it,
61:31 but then one of those is so pseudo universe achieve development that might actually
61:37 something that is useful. So instead dividing by zero, your best
61:45 Don't try to approximate 1/0, making instead of something close to. So
61:53 some point when these large numbers gets , then the zero. And
62:01 But instead of one. And I , well zero. So now they
62:05 a reduced problem. And when you the one of the segments until you
62:10 the so much they're acceptable. So you have another. So this is
62:15 observation in verse and it turns out this is something that can work for
62:22 . So this is they're not directly the inverse if the singularity that composition
62:29 then you have your recent news um money versus trivial just can't work that
62:35 until it gets too dark. And kind of the inverse is useful.
62:42 it was so All right. It's a bit messy and a few more
62:54 . Mm hmm. And get to point. So he goes back to
63:01 terms of the of the composition this already talked about. Ah So we
63:09 that this is among the problems that or others. Yeah, vectors and
63:19 around you normalized inspectors they succeed gonna and so on. This is the
63:27 doctor and scaling the normalization. So is not normal just after new
63:37 the investigation. You have to the of according to interest studies in this
63:46 for example then Yeah. Oh seven. The number of correspondent
63:56 Um but basically you have a number them. They are living in the
64:06 because these things are. Yeah. then yeah You find this as young
64:16 a diagonal correspondent zero wow. And values on the M. T.
64:24 . And then again that's small. or less than we can equal.
64:31 when they use some attention that these he's coming from government instead of a
64:40 . And then um now we do machinery had this on the previous
64:45 So look at this entity eight times this year was under Jerry Times.
64:53 the agreement defines what this guy since was normalized instead of The TV is
65:03 . Some of very much is again it's one of those items values of
65:11 time is the square of the singular is okay so sigma is in this
65:18 of Okay, never. And I'm to little manipulation of these expressions
65:29 Um because the properties uh sigma's define the two products is this guy here
65:44 that's just the simba. So that think kind of the square root of
65:49 thing. Skin orange for the statement then you have what you was defined
66:01 being there. Yeah normalized. So have the same comments here figured
66:09 And then so you have this expression plug it in here, eventually what
66:17 fact that this this expression is. and then that's gonna mean jihad in
66:28 C. So you cannot supply this from the right he is of the
66:33 matrix. what's left on the right side is A and on the left
66:37 side to reduce stigma canceling transit celeste derive it. It's not just computational
66:46 , whatever the government. I went it some example that was doing that
66:55 and I know the retrospective rains So this is An example of them
67:02 the book with anyone treat by two . So here is a three by
67:10 . We started with a T a three by two matrix. He has
67:17 binding values and this is just um your computer uses the services not the
67:26 . The computer shows the properties. get started appearing on similar zero.
67:36 that's some method harbor These are the Eigen vectors to these two. I
67:44 values. And when you try to these two guys and I normalized you
67:55 find out that it's still normalized scenarios their spirit form sticking these two vectors
68:04 into matrix they want and they to one of us went to and you
68:09 track the first one military hearing So it's supposed turning this into a
68:14 vector, there are the same The next one has the opposite
68:19 So that's good to see your 1st 2nd months. So just plugging it
68:24 and then we want to be the reviews and then so there is what
68:31 said the book and stability as its STD because this matrix has now cleared
68:40 with and roll that correspond to things Yes, we don't exist or a
68:47 because the 88 is a two x . So it's Can't handle one or
68:53 single running on Sarah for them. go through the process and finding the
69:01 just from the definition of defined you be You get the U. one
69:06 U two straightforward since they were based the correspondent. These are your
69:11 But there's no fee. The problem it was on the two. I
69:18 . So then again, if you to do the full method and to
69:22 another common, that's what they That's fine. Or that something else
69:29 about. So this is kind of it up with an extra column corresponding
69:34 the extra those signals. Mm The United States. So then and
69:48 what The other day that they stay two. Now it's a full matrix
69:52 filled up with an extra column. is continued and practical being useless because
69:58 want to justify. So it's kind does it or nothing. Um,
70:03 then there was to be that everything comes from your dress vectors of.
70:11 you. Thank you. Thank you aware about this column. And that
70:17 a lot. But I think at time there are limits. So this
70:36 come back to probably the first major the class some insights and kind of
70:45 . So we have this condition number that was to norm um, of
70:55 or the ratio between the national, kind of easier to remember.
71:02 and if this condition number is large is a damn city condition and that
71:07 significance. And the rule of thunder you know, depending about what power
71:15 10 but it's part of that's the . So there's a bunch of matrix
71:27 . You want to come back to examples little example that we had the
71:37 and it wasn't an artist science and did it In Place four digits of
71:44 . I was afraid dangers and So case regions I guess there was 1
71:53 have they been afforded is the records this guy. Here we go,
71:57 with different results. So and then service an example of the main condition
72:06 and that's why these things happen and was kind of the matrix but do
72:12 thing and then we'll figure out these Eigen values. One is a
72:16 very small and the other one is . So if you look at the
72:21 between those cars, you can tell it wanted was hold on um,
72:30 and mail. The condition number not yes, largely expected several. That's
72:45 really. He's three of them. this is nothing was correct actually that's
72:51 vendors final -540 and the plus 300 , no, no if you knew
72:59 the condition number was at the you know what to expect and you
73:03 be able to trust anything about. that's why it's again, condition numbers
73:11 very useful then to understand what you trust in terms of have been from
73:17 simple next time I'll talk about algorithms it's just kind of exercise to show
73:32 , you know, it's a little better and be what kind of era
73:36 you and you expect any solutions for sex? And it basically says the
73:43 N. R and D has blown according to the conditions. So,
73:53 that's what this is. So so today trying to point out the similarity
74:03 the singular value decomposition has kind of . Okay, higher value profits.
74:10 I used them to the characterization of matrix order competition. That is more
74:17 than the gender composition. That is sorry, more properties or the matrix
74:25 we can do for animators and then coming up on the singular values.
74:35 the practice. What is the dimensionality the space and interact with. So
74:44 know, one day single riders get enough relative to the other ones and
74:50 enough to just north and reduce the and take the job. It doesn't
74:57 the final solution. So nice. then next time I thought about how
75:06 breakfast actually conduct singular values and the . I love that. So,
75:20 . Thank you. Yeah, Thank . Thank you. Mm hmm.
75:32 you. He's a good
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